: Direct application of Theorem \(5.14\)  of [1]} guarantees the existence of a unitary \({S}\)  such that
\(\begin{aligned}\mathrm {LHS}=&\mathbb {E}\Vert {U} {\Sigma }^{1 / 2}-\tilde{{U}} \tilde{\Sigma }^{1 / 2} {S}\Vert _{F}^{2}+\mathbb {E}\Vert V {\Sigma }^{1 / 2}-\tilde{{V}} \tilde{\Sigma }^{1 / 2} {S}\Vert _{F}^{2} \\&\le \frac{2}{\sqrt{2}-1} \frac{\mathbb {E}\Vert {L}-\tilde{{L}}\Vert _{F}^{2}}{\sigma _{\min }({L})},\end{aligned}\) 
